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In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−√3,−1)
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.43
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.43Capitolo 5, Problema 5.43
In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.
A hyperbola: x = h + a sec t, y = k + b tan t
Guida verificata passo dopo passo1
Start with the given parametric equations of the hyperbola: \(x = h + a \sec t\) and \(y = k + b \tan t\).
Recall the fundamental trigonometric identity relating secant and tangent: \(\sec^2 t - \tan^2 t = 1\).
Express \(\sec t\) and \(\tan t\) in terms of \(x\) and \(y\) by isolating them from the parametric equations: \(\sec t = \frac{x - h}{a}\) and \(\tan t = \frac{y - k}{b}\).
Substitute these expressions into the identity \(\sec^2 t - \tan^2 t = 1\) to eliminate the parameter \(t\).
Simplify the resulting equation to write it in the standard form of a hyperbola: \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how to manipulate these equations is essential for eliminating the parameter and finding a direct relationship between x and y.
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Percorso guidato
08:02
Parameterizing Equations
Trigonometric Identities Involving Secant and Tangent
The identity sec²t - tan²t = 1 is crucial when working with parametric equations involving sec t and tan t. This identity allows the elimination of the parameter t by relating sec t and tan t algebraically.
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Percorso guidato
5:32Fundamental Trigonometric Identities
Standard Form of a Hyperbola
The standard form of a hyperbola centered at (h, k) is ((x - h)² / a²) - ((y - k)² / b²) = 1. Recognizing this form helps in rewriting the equation after eliminating the parameter to identify the conic section clearly.
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Percorso guidato
05:50Drawing Angles in Standard Position
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